LMFDB - Embedded newform 637.2.a.j.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [637,2,Mod(1,637)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("637.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(637, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 637 = 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	637.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,1,2,3,-2,-4,0,3,7,8,2,12,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$5.08647060876$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	637.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.470683 q^{2} -2.24914 q^{3} -1.77846 q^{4} -0.529317 q^{5} -1.05863 q^{6} -1.77846 q^{8} +2.05863 q^{9} -0.249141 q^{10} -2.24914 q^{11} +4.00000 q^{12} -1.00000 q^{13} +1.19051 q^{15} +2.71982 q^{16} +1.30777 q^{17} +0.968964 q^{18} +1.47068 q^{19} +0.941367 q^{20} -1.05863 q^{22} +5.83709 q^{23} +4.00000 q^{24} -4.71982 q^{25} -0.470683 q^{26} +2.11727 q^{27} +5.22154 q^{29} +0.560352 q^{30} +7.02760 q^{31} +4.83709 q^{32} +5.05863 q^{33} +0.615547 q^{34} -3.66119 q^{36} -2.36641 q^{37} +0.692226 q^{38} +2.24914 q^{39} +0.941367 q^{40} -6.49828 q^{41} +11.3940 q^{43} +4.00000 q^{44} -1.08967 q^{45} +2.74742 q^{46} -8.58451 q^{47} -6.11727 q^{48} -2.22154 q^{50} -2.94137 q^{51} +1.77846 q^{52} +11.2767 q^{53} +0.996562 q^{54} +1.19051 q^{55} -3.30777 q^{57} +2.45769 q^{58} +12.1725 q^{59} -2.11727 q^{60} +2.00000 q^{61} +3.30777 q^{62} -3.16291 q^{64} +0.529317 q^{65} +2.38101 q^{66} -15.9379 q^{67} -2.32582 q^{68} -13.1284 q^{69} +1.19051 q^{71} -3.66119 q^{72} -7.64315 q^{73} -1.11383 q^{74} +10.6155 q^{75} -2.61555 q^{76} +1.05863 q^{78} -1.33881 q^{79} -1.43965 q^{80} -10.9379 q^{81} -3.05863 q^{82} +16.3500 q^{83} -0.692226 q^{85} +5.36297 q^{86} -11.7440 q^{87} +4.00000 q^{88} -6.91033 q^{89} -0.512889 q^{90} -10.3810 q^{92} -15.8061 q^{93} -4.04059 q^{94} -0.778457 q^{95} -10.8793 q^{96} +3.47068 q^{97} -4.63016 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 2 q^{3} + 3 q^{4} - 2 q^{5} - 4 q^{6} + 3 q^{8} + 7 q^{9} + 8 q^{10} + 2 q^{11} + 12 q^{12} - 3 q^{13} - 6 q^{15} - q^{16} - 4 q^{17} - 15 q^{18} + 4 q^{19} + 2 q^{20} - 4 q^{22} + 10 q^{23}+ \cdots + 14 q^{99}+O(q^{100}) $ 3 * q + q^2 + 2 * q^3 + 3 * q^4 - 2 * q^5 - 4 * q^6 + 3 * q^8 + 7 * q^9 + 8 * q^10 + 2 * q^11 + 12 * q^12 - 3 * q^13 - 6 * q^15 - q^16 - 4 * q^17 - 15 * q^18 + 4 * q^19 + 2 * q^20 - 4 * q^22 + 10 * q^23 + 12 * q^24 - 5 * q^25 - q^26 + 8 * q^27 + 24 * q^29 + 20 * q^30 + 4 * q^31 + 7 * q^32 + 16 * q^33 - 14 * q^34 - q^36 + 10 * q^38 - 2 * q^39 + 2 * q^40 - 2 * q^41 + 10 * q^43 + 12 * q^44 - 22 * q^45 - 18 * q^46 + 8 * q^47 - 20 * q^48 - 15 * q^50 - 8 * q^51 - 3 * q^52 + 8 * q^53 - 32 * q^54 - 6 * q^55 - 2 * q^57 + 12 * q^58 + 4 * q^59 - 8 * q^60 + 6 * q^61 + 2 * q^62 - 17 * q^64 + 2 * q^65 - 12 * q^66 - 12 * q^67 - 22 * q^68 + 6 * q^69 - 6 * q^71 - q^72 + 10 * q^73 + 30 * q^74 + 16 * q^75 + 8 * q^76 + 4 * q^78 - 14 * q^79 + 14 * q^80 + 3 * q^81 - 10 * q^82 + 12 * q^83 - 10 * q^85 - 26 * q^86 + 26 * q^87 + 12 * q^88 - 2 * q^89 + 28 * q^90 - 12 * q^92 - 22 * q^93 + 10 * q^94 + 6 * q^95 + 4 * q^96 + 10 * q^97 + 14 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.470683	0.332823	0.166412	−	0.986056i	$-0.446782\pi$
$2$	0.470683	0.332823	0.166412	+	0.986056i	$0.446782\pi$
$3$	−2.24914	−1.29854	−0.649271	−	0.760557i	$-0.724926\pi$
$3$	−2.24914	−1.29854	−0.649271	+	0.760557i	$0.724926\pi$
$4$	−1.77846	−0.889229
$5$	−0.529317	−0.236718	−0.118359	−	0.992971i	$-0.537763\pi$
$5$	−0.529317	−0.236718	−0.118359	+	0.992971i	$0.537763\pi$
$6$	−1.05863	−0.432185
$7$	0	0
$7$	0	0
$8$	−1.77846	−0.628780
$9$	2.05863	0.686211
$10$	−0.249141	−0.0787852
$11$	−2.24914	−0.678141	−0.339071	−	0.940761i	$-0.610113\pi$
$11$	−2.24914	−0.678141	−0.339071	+	0.940761i	$0.610113\pi$
$12$	4.00000	1.15470
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.19051	0.307388
$16$	2.71982	0.679956
$17$	1.30777	0.317182	0.158591	−	0.987344i	$-0.449305\pi$
$17$	1.30777	0.317182	0.158591	+	0.987344i	$0.449305\pi$
$18$	0.968964	0.228387
$19$	1.47068	0.337398	0.168699	−	0.985668i	$-0.446043\pi$
$19$	1.47068	0.337398	0.168699	+	0.985668i	$0.446043\pi$
$20$	0.941367	0.210496
$21$	0	0
$22$	−1.05863	−0.225701
$23$	5.83709	1.21712	0.608559	−	0.793509i	$-0.291748\pi$
$23$	5.83709	1.21712	0.608559	+	0.793509i	$0.291748\pi$
$24$	4.00000	0.816497
$25$	−4.71982	−0.943965
$26$	−0.470683	−0.0923086
$27$	2.11727	0.407468
$28$	0	0
$29$	5.22154	0.969616	0.484808	−	0.874621i	$-0.338889\pi$
$29$	5.22154	0.969616	0.484808	+	0.874621i	$0.338889\pi$
$30$	0.560352	0.102306
$31$	7.02760	1.26219	0.631097	−	0.775704i	$-0.282605\pi$
$31$	7.02760	1.26219	0.631097	+	0.775704i	$0.282605\pi$
$32$	4.83709	0.855085
$33$	5.05863	0.880595
$34$	0.615547	0.105566
$35$	0	0
$36$	−3.66119	−0.610198
$37$	−2.36641	−0.389035	−0.194517	−	0.980899i	$-0.562314\pi$
$37$	−2.36641	−0.389035	−0.194517	+	0.980899i	$0.562314\pi$
$38$	0.692226	0.112294
$39$	2.24914	0.360151
$40$	0.941367	0.148843
$41$	−6.49828	−1.01486	−0.507431	−	0.861693i	$-0.669405\pi$
$41$	−6.49828	−1.01486	−0.507431	+	0.861693i	$0.669405\pi$
$42$	0	0
$43$	11.3940	1.73757	0.868785	−	0.495190i	$-0.164902\pi$
$43$	11.3940	1.73757	0.868785	+	0.495190i	$0.164902\pi$
$44$	4.00000	0.603023
$45$	−1.08967	−0.162438
$46$	2.74742	0.405085
$47$	−8.58451	−1.25218	−0.626090	−	0.779751i	$-0.715346\pi$
$47$	−8.58451	−1.25218	−0.626090	+	0.779751i	$0.715346\pi$
$48$	−6.11727	−0.882951
$49$	0	0
$50$	−2.22154	−0.314174
$51$	−2.94137	−0.411874
$52$	1.77846	0.246628
$53$	11.2767	1.54898	0.774490	−	0.632587i	$-0.218007\pi$
$53$	11.2767	1.54898	0.774490	+	0.632587i	$0.218007\pi$
$54$	0.996562	0.135615
$55$	1.19051	0.160528
$56$	0	0
$57$	−3.30777	−0.438125
$58$	2.45769	0.322711
$59$	12.1725	1.58472	0.792360	−	0.610054i	$-0.208852\pi$
$59$	12.1725	1.58472	0.792360	+	0.610054i	$0.208852\pi$
$60$	−2.11727	−0.273338
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	3.30777	0.420088
$63$	0	0
$64$	−3.16291	−0.395364
$65$	0.529317	0.0656536
$66$	2.38101	0.293083
$67$	−15.9379	−1.94713	−0.973564	−	0.228415i	$-0.926646\pi$
$67$	−15.9379	−1.94713	−0.973564	+	0.228415i	$0.926646\pi$
$68$	−2.32582	−0.282047
$69$	−13.1284	−1.58048
$70$	0	0
$71$	1.19051	0.141287	0.0706436	−	0.997502i	$-0.477495\pi$
$71$	1.19051	0.141287	0.0706436	+	0.997502i	$0.477495\pi$
$72$	−3.66119	−0.431475
$73$	−7.64315	−0.894562	−0.447281	−	0.894393i	$-0.647608\pi$
$73$	−7.64315	−0.894562	−0.447281	+	0.894393i	$0.647608\pi$
$74$	−1.11383	−0.129480
$75$	10.6155	1.22578
$76$	−2.61555	−0.300024
$77$	0	0
$78$	1.05863	0.119867
$79$	−1.33881	−0.150628	−0.0753139	−	0.997160i	$-0.523996\pi$
$79$	−1.33881	−0.150628	−0.0753139	+	0.997160i	$0.523996\pi$
$80$	−1.43965	−0.160958
$81$	−10.9379	−1.21533
$82$	−3.05863	−0.337770
$83$	16.3500	1.79464	0.897322	−	0.441377i	$-0.145510\pi$
$83$	16.3500	1.79464	0.897322	+	0.441377i	$0.145510\pi$
$84$	0	0
$85$	−0.692226	−0.0750825
$86$	5.36297	0.578304
$87$	−11.7440	−1.25909
$88$	4.00000	0.426401
$89$	−6.91033	−0.732494	−0.366247	−	0.930518i	$-0.619357\pi$
$89$	−6.91033	−0.732494	−0.366247	+	0.930518i	$0.619357\pi$
$90$	−0.512889	−0.0540632
$91$	0	0
$92$	−10.3810	−1.08230
$93$	−15.8061	−1.63901
$94$	−4.04059	−0.416755
$95$	−0.778457	−0.0798680
$96$	−10.8793	−1.11036
$97$	3.47068	0.352395	0.176197	−	0.984355i	$-0.443620\pi$
$97$	3.47068	0.352395	0.176197	+	0.984355i	$0.443620\pi$
$98$	0	0
$99$	−4.63016	−0.465348
$100$	8.39400	0.839400
$101$	−7.75086	−0.771239	−0.385620	−	0.922658i	$-0.626012\pi$
$101$	−7.75086	−0.771239	−0.385620	+	0.922658i	$0.626012\pi$
$102$	−1.38445	−0.137081
$103$	16.9966	1.67472	0.837361	−	0.546651i	$-0.184098\pi$
$103$	16.9966	1.67472	0.837361	+	0.546651i	$0.184098\pi$
$104$	1.77846	0.174392
$105$	0	0
$106$	5.30777	0.515537
$107$	−5.55691	−0.537207	−0.268604	−	0.963251i	$-0.586562\pi$
$107$	−5.55691	−0.537207	−0.268604	+	0.963251i	$0.586562\pi$
$108$	−3.76547	−0.362332
$109$	7.92332	0.758917	0.379458	−	0.925209i	$-0.376110\pi$
$109$	7.92332	0.758917	0.379458	+	0.925209i	$0.376110\pi$
$110$	0.560352	0.0534275
$111$	5.32238	0.505178
$112$	0	0
$113$	−9.89229	−0.930588	−0.465294	−	0.885156i	$-0.654051\pi$
$113$	−9.89229	−0.930588	−0.465294	+	0.885156i	$0.654051\pi$
$114$	−1.55691	−0.145818
$115$	−3.08967	−0.288113
$116$	−9.28629	−0.862210
$117$	−2.05863	−0.190321
$118$	5.72938	0.527432
$119$	0	0
$120$	−2.11727	−0.193279
$121$	−5.94137	−0.540124
$122$	0.941367	0.0852273
$123$	14.6155	1.31784
$124$	−12.4983	−1.12238
$125$	5.14486	0.460171
$126$	0	0
$127$	0.824101	0.0731271	0.0365635	−	0.999331i	$-0.488359\pi$
$127$	0.824101	0.0731271	0.0365635	+	0.999331i	$0.488359\pi$
$128$	−11.1629	−0.986671
$129$	−25.6267	−2.25631
$130$	0.249141	0.0218511
$131$	−10.6155	−0.927485	−0.463742	−	0.885970i	$-0.653494\pi$
$131$	−10.6155	−0.927485	−0.463742	+	0.885970i	$0.653494\pi$
$132$	−8.99656	−0.783050
$133$	0	0
$134$	−7.50172	−0.648050
$135$	−1.12070	−0.0964549
$136$	−2.32582	−0.199437
$137$	11.3630	0.970804	0.485402	−	0.874291i	$-0.338673\pi$
$137$	11.3630	0.970804	0.485402	+	0.874291i	$0.338673\pi$
$138$	−6.17934	−0.526020
$139$	13.9233	1.18096	0.590480	−	0.807052i	$-0.298938\pi$
$139$	13.9233	1.18096	0.590480	+	0.807052i	$0.298938\pi$
$140$	0	0
$141$	19.3078	1.62601
$142$	0.560352	0.0470237
$143$	2.24914	0.188083
$144$	5.59912	0.466593
$145$	−2.76385	−0.229525
$146$	−3.59750	−0.297731
$147$	0	0
$148$	4.20855	0.345941
$149$	9.30777	0.762523	0.381261	−	0.924467i	$-0.375490\pi$
$149$	9.30777	0.762523	0.381261	+	0.924467i	$0.375490\pi$
$150$	4.99656	0.407968
$151$	7.07324	0.575612	0.287806	−	0.957689i	$-0.407074\pi$
$151$	7.07324	0.575612	0.287806	+	0.957689i	$0.407074\pi$
$152$	−2.61555	−0.212149
$153$	2.69223	0.217654
$154$	0	0
$155$	−3.71982	−0.298783
$156$	−4.00000	−0.320256
$157$	6.04059	0.482091	0.241046	−	0.970514i	$-0.422510\pi$
$157$	6.04059	0.482091	0.241046	+	0.970514i	$0.422510\pi$
$158$	−0.630155	−0.0501325
$159$	−25.3630	−2.01141
$160$	−2.56035	−0.202414
$161$	0	0
$162$	−5.14830	−0.404489
$163$	6.38101	0.499800	0.249900	−	0.968272i	$-0.419602\pi$
$163$	6.38101	0.499800	0.249900	+	0.968272i	$0.419602\pi$
$164$	11.5569	0.902443
$165$	−2.67762	−0.208452
$166$	7.69566	0.597299
$167$	16.5845	1.28335	0.641674	−	0.766977i	$-0.278240\pi$
$167$	16.5845	1.28335	0.641674	+	0.766977i	$0.278240\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−0.325819	−0.0249892
$171$	3.02760	0.231526
$172$	−20.2637	−1.54510
$173$	23.3009	1.77153	0.885767	−	0.464130i	$-0.153633\pi$
$173$	23.3009	1.77153	0.885767	+	0.464130i	$0.153633\pi$
$174$	−5.52770	−0.419054
$175$	0	0
$176$	−6.11727	−0.461106
$177$	−27.3776	−2.05782
$178$	−3.25258	−0.243791
$179$	21.0422	1.57277	0.786384	−	0.617738i	$-0.211951\pi$
$179$	21.0422	1.57277	0.786384	+	0.617738i	$0.211951\pi$
$180$	1.93793	0.144445
$181$	−16.7474	−1.24483	−0.622413	−	0.782689i	$-0.713848\pi$
$181$	−16.7474	−1.24483	−0.622413	+	0.782689i	$0.713848\pi$
$182$	0	0
$183$	−4.49828	−0.332523
$184$	−10.3810	−0.765299
$185$	1.25258	0.0920914
$186$	−7.43965	−0.545501
$187$	−2.94137	−0.215094
$188$	15.2672	1.11347
$189$	0	0
$190$	−0.366407	−0.0265819
$191$	−7.43965	−0.538314	−0.269157	−	0.963096i	$-0.586745\pi$
$191$	−7.43965	−0.538314	−0.269157	+	0.963096i	$0.586745\pi$
$192$	7.11383	0.513396
$193$	−1.50172	−0.108096	−0.0540480	−	0.998538i	$-0.517212\pi$
$193$	−1.50172	−0.108096	−0.0540480	+	0.998538i	$0.517212\pi$
$194$	1.63359	0.117285
$195$	−1.19051	−0.0852540
$196$	0	0
$197$	23.9931	1.70944	0.854720	−	0.519090i	$-0.173729\pi$
$197$	23.9931	1.70944	0.854720	+	0.519090i	$0.173729\pi$
$198$	−2.17934	−0.154879
$199$	2.01461	0.142812	0.0714059	−	0.997447i	$-0.477251\pi$
$199$	2.01461	0.142812	0.0714059	+	0.997447i	$0.477251\pi$
$200$	8.39400	0.593546
$201$	35.8466	2.52843
$202$	−3.64820	−0.256687
$203$	0	0
$204$	5.23109	0.366250
$205$	3.43965	0.240235
$206$	8.00000	0.557386
$207$	12.0164	0.835199
$208$	−2.71982	−0.188586
$209$	−3.30777	−0.228803
$210$	0	0
$211$	10.1008	0.695370	0.347685	−	0.937611i	$-0.386968\pi$
$211$	10.1008	0.695370	0.347685	+	0.937611i	$0.386968\pi$
$212$	−20.0552	−1.37740
$213$	−2.67762	−0.183467
$214$	−2.61555	−0.178795
$215$	−6.03104	−0.411313
$216$	−3.76547	−0.256208
$217$	0	0
$218$	3.72938	0.252585
$219$	17.1905	1.16163
$220$	−2.11727	−0.142746
$221$	−1.30777	−0.0879704
$222$	2.50516	0.168135
$223$	−10.1414	−0.679120	−0.339560	−	0.940584i	$-0.610278\pi$
$223$	−10.1414	−0.679120	−0.339560	+	0.940584i	$0.610278\pi$
$224$	0	0
$225$	−9.71639	−0.647759
$226$	−4.65613	−0.309721
$227$	5.38445	0.357379	0.178689	−	0.983906i	$-0.442814\pi$
$227$	5.38445	0.357379	0.178689	+	0.983906i	$0.442814\pi$
$228$	5.88273	0.389594
$229$	−3.32238	−0.219549	−0.109775	−	0.993957i	$-0.535013\pi$
$229$	−3.32238	−0.219549	−0.109775	+	0.993957i	$0.535013\pi$
$230$	−1.45426	−0.0958908
$231$	0	0
$232$	−9.28629	−0.609675
$233$	13.7198	0.898816	0.449408	−	0.893327i	$-0.351635\pi$
$233$	13.7198	0.898816	0.449408	+	0.893327i	$0.351635\pi$
$234$	−0.968964	−0.0633432
$235$	4.54392	0.296413
$236$	−21.6482	−1.40918
$237$	3.01117	0.195597
$238$	0	0
$239$	−3.50172	−0.226507	−0.113254	−	0.993566i	$-0.536127\pi$
$239$	−3.50172	−0.226507	−0.113254	+	0.993566i	$0.536127\pi$
$240$	3.23797	0.209010
$241$	−1.58795	−0.102289	−0.0511444	−	0.998691i	$-0.516287\pi$
$241$	−1.58795	−0.102289	−0.0511444	+	0.998691i	$0.516287\pi$
$242$	−2.79650	−0.179766
$243$	18.2491	1.17068
$244$	−3.55691	−0.227708
$245$	0	0
$246$	6.87930	0.438608
$247$	−1.47068	−0.0935773
$248$	−12.4983	−0.793642
$249$	−36.7734	−2.33042
$250$	2.42160	0.153156
$251$	4.92676	0.310974	0.155487	−	0.987838i	$-0.450305\pi$
$251$	4.92676	0.310974	0.155487	+	0.987838i	$0.450305\pi$
$252$	0	0
$253$	−13.1284	−0.825378
$254$	0.387890	0.0243384
$255$	1.55691	0.0974978
$256$	1.07162	0.0669764
$257$	8.01461	0.499938	0.249969	−	0.968254i	$-0.419580\pi$
$257$	8.01461	0.499938	0.249969	+	0.968254i	$0.419580\pi$
$258$	−12.0621	−0.750952
$259$	0	0
$260$	−0.941367	−0.0583811
$261$	10.7492	0.665361
$262$	−4.99656	−0.308689
$263$	1.60256	0.0988179	0.0494090	−	0.998779i	$-0.484266\pi$
$263$	1.60256	0.0988179	0.0494090	+	0.998779i	$0.484266\pi$
$264$	−8.99656	−0.553700
$265$	−5.96896	−0.366671
$266$	0	0
$267$	15.5423	0.951174
$268$	28.3449	1.73144
$269$	11.8207	0.720719	0.360359	−	0.932814i	$-0.382654\pi$
$269$	11.8207	0.720719	0.360359	+	0.932814i	$0.382654\pi$
$270$	−0.527497	−0.0321024
$271$	−21.8827	−1.32928	−0.664641	−	0.747163i	$-0.731415\pi$
$271$	−21.8827	−1.32928	−0.664641	+	0.747163i	$0.731415\pi$
$272$	3.55691	0.215670
$273$	0	0
$274$	5.34836	0.323106
$275$	10.6155	0.640142
$276$	23.3484	1.40541
$277$	−10.2181	−0.613946	−0.306973	−	0.951718i	$-0.599316\pi$
$277$	−10.2181	−0.613946	−0.306973	+	0.951718i	$0.599316\pi$
$278$	6.55348	0.393051
$279$	14.4672	0.866131
$280$	0	0
$281$	1.54231	0.0920063	0.0460031	−	0.998941i	$-0.485352\pi$
$281$	1.54231	0.0920063	0.0460031	+	0.998941i	$0.485352\pi$
$282$	9.08785	0.541174
$283$	−15.8466	−0.941985	−0.470993	−	0.882137i	$-0.656104\pi$
$283$	−15.8466	−0.941985	−0.470993	+	0.882137i	$0.656104\pi$
$284$	−2.11727	−0.125637
$285$	1.75086	0.103712
$286$	1.05863	0.0625983
$287$	0	0
$288$	9.95779	0.586769
$289$	−15.2897	−0.899396
$290$	−1.30090	−0.0763914
$291$	−7.80605	−0.457599
$292$	13.5930	0.795470
$293$	−11.0828	−0.647464	−0.323732	−	0.946149i	$-0.604938\pi$
$293$	−11.0828	−0.647464	−0.323732	+	0.946149i	$0.604938\pi$
$294$	0	0
$295$	−6.44309	−0.375131
$296$	4.20855	0.244617
$297$	−4.76203	−0.276321
$298$	4.38101	0.253785
$299$	−5.83709	−0.337568
$300$	−18.8793	−1.09000
$301$	0	0
$302$	3.32926	0.191577
$303$	17.4328	1.00149
$304$	4.00000	0.229416
$305$	−1.05863	−0.0606172
$306$	1.26719	0.0724402
$307$	−20.4121	−1.16498	−0.582489	−	0.812839i	$-0.697921\pi$
$307$	−20.4121	−1.16498	−0.582489	+	0.812839i	$0.697921\pi$
$308$	0	0
$309$	−38.2277	−2.17470
$310$	−1.75086	−0.0994421
$311$	1.92332	0.109062	0.0545308	−	0.998512i	$-0.482634\pi$
$311$	1.92332	0.109062	0.0545308	+	0.998512i	$0.482634\pi$
$312$	−4.00000	−0.226455
$313$	14.3664	0.812037	0.406019	−	0.913865i	$-0.366917\pi$
$313$	14.3664	0.812037	0.406019	+	0.913865i	$0.366917\pi$
$314$	2.84320	0.160451
$315$	0	0
$316$	2.38101	0.133943
$317$	15.5569	0.873763	0.436882	−	0.899519i	$-0.356083\pi$
$317$	15.5569	0.873763	0.436882	+	0.899519i	$0.356083\pi$
$318$	−11.9379	−0.669446
$319$	−11.7440	−0.657537
$320$	1.67418	0.0935895
$321$	12.4983	0.697586
$322$	0	0
$323$	1.92332	0.107016
$324$	19.4526	1.08070
$325$	4.71982	0.261809
$326$	3.00344	0.166345
$327$	−17.8207	−0.985485
$328$	11.5569	0.638124
$329$	0	0
$330$	−1.26031	−0.0693778
$331$	31.5500	1.73415	0.867073	−	0.498180i	$-0.165998\pi$
$331$	31.5500	1.73415	0.867073	+	0.498180i	$0.165998\pi$
$332$	−29.0777	−1.59585
$333$	−4.87156	−0.266960
$334$	7.80605	0.427128
$335$	8.43621	0.460919
$336$	0	0
$337$	−8.42666	−0.459029	−0.229515	−	0.973305i	$-0.573714\pi$
$337$	−8.42666	−0.459029	−0.229515	+	0.973305i	$0.573714\pi$
$338$	0.470683	0.0256018
$339$	22.2491	1.20841
$340$	1.23109	0.0667655
$341$	−15.8061	−0.855946
$342$	1.42504	0.0770573
$343$	0	0
$344$	−20.2637	−1.09255
$345$	6.94910	0.374127
$346$	10.9673	0.589608
$347$	19.3484	1.03867	0.519337	−	0.854569i	$-0.326179\pi$
$347$	19.3484	1.03867	0.519337	+	0.854569i	$0.326179\pi$
$348$	20.8862	1.11962
$349$	−27.2553	−1.45894	−0.729470	−	0.684013i	$-0.760233\pi$
$349$	−27.2553	−1.45894	−0.729470	+	0.684013i	$0.760233\pi$
$350$	0	0
$351$	−2.11727	−0.113011
$352$	−10.8793	−0.579868
$353$	25.6742	1.36650	0.683249	−	0.730185i	$-0.260566\pi$
$353$	25.6742	1.36650	0.683249	+	0.730185i	$0.260566\pi$
$354$	−12.8862	−0.684892
$355$	−0.630155	−0.0334452
$356$	12.2897	0.651354
$357$	0	0
$358$	9.90422	0.523454
$359$	−23.4182	−1.23596	−0.617982	−	0.786192i	$-0.712049\pi$
$359$	−23.4182	−1.23596	−0.617982	+	0.786192i	$0.712049\pi$
$360$	1.93793	0.102138
$361$	−16.8371	−0.886163
$362$	−7.88273	−0.414307
$363$	13.3630	0.701374
$364$	0	0
$365$	4.04564	0.211759
$366$	−2.11727	−0.110671
$367$	14.6854	0.766569	0.383285	−	0.923630i	$-0.374793\pi$
$367$	14.6854	0.766569	0.383285	+	0.923630i	$0.374793\pi$
$368$	15.8759	0.827586
$369$	−13.3776	−0.696409
$370$	0.589568	0.0306502
$371$	0	0
$372$	28.1104	1.45746
$373$	−23.6673	−1.22545	−0.612723	−	0.790298i	$-0.709926\pi$
$373$	−23.6673	−1.22545	−0.612723	+	0.790298i	$0.709926\pi$
$374$	−1.38445	−0.0715883
$375$	−11.5715	−0.597551
$376$	15.2672	0.787345
$377$	−5.22154	−0.268923
$378$	0	0
$379$	−32.7405	−1.68177	−0.840884	−	0.541215i	$-0.817965\pi$
$379$	−32.7405	−1.68177	−0.840884	+	0.541215i	$0.817965\pi$
$380$	1.38445	0.0710209
$381$	−1.85352	−0.0949586
$382$	−3.50172	−0.179164
$383$	22.6155	1.15560	0.577800	−	0.816178i	$-0.303911\pi$
$383$	22.6155	1.15560	0.577800	+	0.816178i	$0.303911\pi$
$384$	25.1070	1.28123
$385$	0	0
$386$	−0.706834	−0.0359769
$387$	23.4561	1.19234
$388$	−6.17246	−0.313359
$389$	38.0483	1.92913	0.964563	−	0.263852i	$-0.0849930\pi$
$389$	38.0483	1.92913	0.964563	+	0.263852i	$0.0849930\pi$
$390$	−0.560352	−0.0283745
$391$	7.63359	0.386047
$392$	0	0
$393$	23.8759	1.20438
$394$	11.2932	0.568941
$395$	0.708654	0.0356562
$396$	8.23453	0.413801
$397$	11.7052	0.587468	0.293734	−	0.955887i	$-0.405102\pi$
$397$	11.7052	0.587468	0.293734	+	0.955887i	$0.405102\pi$
$398$	0.948243	0.0475311
$399$	0	0
$400$	−12.8371	−0.641855
$401$	3.55691	0.177624	0.0888119	−	0.996048i	$-0.471693\pi$
$401$	3.55691	0.177624	0.0888119	+	0.996048i	$0.471693\pi$
$402$	16.8724	0.841520
$403$	−7.02760	−0.350070
$404$	13.7846	0.685808
$405$	5.78963	0.287689
$406$	0	0
$407$	5.32238	0.263821
$408$	5.23109	0.258978
$409$	−5.26213	−0.260196	−0.130098	−	0.991501i	$-0.541529\pi$
$409$	−5.26213	−0.260196	−0.130098	+	0.991501i	$0.541529\pi$
$410$	1.61899	0.0799560
$411$	−25.5569	−1.26063
$412$	−30.2277	−1.48921
$413$	0	0
$414$	5.65593	0.277974
$415$	−8.65432	−0.424824
$416$	−4.83709	−0.237158
$417$	−31.3155	−1.53353
$418$	−1.55691	−0.0761512
$419$	−26.0337	−1.27183	−0.635915	−	0.771759i	$-0.719377\pi$
$419$	−26.0337	−1.27183	−0.635915	+	0.771759i	$0.719377\pi$
$420$	0	0
$421$	22.2423	1.08402	0.542011	−	0.840372i	$-0.317663\pi$
$421$	22.2423	1.08402	0.542011	+	0.840372i	$0.317663\pi$
$422$	4.75430	0.231436
$423$	−17.6724	−0.859260
$424$	−20.0552	−0.973966
$425$	−6.17246	−0.299408
$426$	−1.26031	−0.0610622
$427$	0	0
$428$	9.88273	0.477700
$429$	−5.05863	−0.244233
$430$	−2.83871	−0.136895
$431$	27.6742	1.33302	0.666509	−	0.745497i	$-0.267788\pi$
$431$	27.6742	1.33302	0.666509	+	0.745497i	$0.267788\pi$
$432$	5.75859	0.277060
$433$	12.7880	0.614552	0.307276	−	0.951620i	$-0.400582\pi$
$433$	12.7880	0.614552	0.307276	+	0.951620i	$0.400582\pi$
$434$	0	0
$435$	6.21629	0.298048
$436$	−14.0913	−0.674850
$437$	8.58451	0.410653
$438$	8.09129	0.386617
$439$	18.1656	0.866996	0.433498	−	0.901155i	$-0.357279\pi$
$439$	18.1656	0.866996	0.433498	+	0.901155i	$0.357279\pi$
$440$	−2.11727	−0.100937
$441$	0	0
$442$	−0.615547	−0.0292786
$443$	0.107714	0.00511767	0.00255883	−	0.999997i	$-0.499185\pi$
$443$	0.107714	0.00511767	0.00255883	+	0.999997i	$0.499185\pi$
$444$	−9.46563	−0.449219
$445$	3.65775	0.173394
$446$	−4.77340	−0.226027
$447$	−20.9345	−0.990167
$448$	0	0
$449$	−22.1725	−1.04638	−0.523192	−	0.852215i	$-0.675259\pi$
$449$	−22.1725	−1.04638	−0.523192	+	0.852215i	$0.675259\pi$
$450$	−4.57334	−0.215589
$451$	14.6155	0.688219
$452$	17.5930	0.827505
$453$	−15.9087	−0.747457
$454$	2.53437	0.118944
$455$	0	0
$456$	5.88273	0.275484
$457$	4.35953	0.203930	0.101965	−	0.994788i	$-0.467487\pi$
$457$	4.35953	0.203930	0.101965	+	0.994788i	$0.467487\pi$
$458$	−1.56379	−0.0730711
$459$	2.76891	0.129241
$460$	5.49484	0.256198
$461$	−32.3810	−1.50813	−0.754067	−	0.656797i	$-0.771911\pi$
$461$	−32.3810	−1.50813	−0.754067	+	0.656797i	$0.771911\pi$
$462$	0	0
$463$	8.36641	0.388820	0.194410	−	0.980920i	$-0.437721\pi$
$463$	8.36641	0.388820	0.194410	+	0.980920i	$0.437721\pi$
$464$	14.2017	0.659296
$465$	8.36641	0.387983
$466$	6.45769	0.299147
$467$	19.5423	0.904310	0.452155	−	0.891939i	$-0.350655\pi$
$467$	19.5423	0.904310	0.452155	+	0.891939i	$0.350655\pi$
$468$	3.66119	0.169239
$469$	0	0
$470$	2.13875	0.0986532
$471$	−13.5861	−0.626016
$472$	−21.6482	−0.996439
$473$	−25.6267	−1.17832
$474$	1.41731	0.0650991
$475$	−6.94137	−0.318492
$476$	0	0
$477$	23.2147	1.06293
$478$	−1.64820	−0.0753870
$479$	28.5224	1.30322	0.651612	−	0.758553i	$-0.274093\pi$
$479$	28.5224	1.30322	0.651612	+	0.758553i	$0.274093\pi$
$480$	5.75859	0.262843
$481$	2.36641	0.107899
$482$	−0.747422	−0.0340441
$483$	0	0
$484$	10.5665	0.480294
$485$	−1.83709	−0.0834180
$486$	8.58957	0.389631
$487$	24.8241	1.12489	0.562444	−	0.826836i	$-0.309861\pi$
$487$	24.8241	1.12489	0.562444	+	0.826836i	$0.309861\pi$
$488$	−3.55691	−0.161014
$489$	−14.3518	−0.649011
$490$	0	0
$491$	29.1690	1.31638	0.658190	−	0.752852i	$-0.271322\pi$
$491$	29.1690	1.31638	0.658190	+	0.752852i	$0.271322\pi$
$492$	−25.9931	−1.17186
$493$	6.82860	0.307545
$494$	−0.692226	−0.0311447
$495$	2.45082	0.110156
$496$	19.1138	0.858236
$497$	0	0
$498$	−17.3086	−0.775618
$499$	−33.3009	−1.49075	−0.745376	−	0.666644i	$-0.767730\pi$
$499$	−33.3009	−1.49075	−0.745376	+	0.666644i	$0.767730\pi$
$500$	−9.14992	−0.409197
$501$	−37.3009	−1.66648
$502$	2.31894	0.103500
$503$	12.3258	0.549581	0.274791	−	0.961504i	$-0.411391\pi$
$503$	12.3258	0.549581	0.274791	+	0.961504i	$0.411391\pi$
$504$	0	0
$505$	4.10266	0.182566
$506$	−6.17934	−0.274705
$507$	−2.24914	−0.0998878
$508$	−1.46563	−0.0650267
$509$	−23.7052	−1.05072	−0.525358	−	0.850882i	$-0.676068\pi$
$509$	−23.7052	−1.05072	−0.525358	+	0.850882i	$0.676068\pi$
$510$	0.732814	0.0324495
$511$	0	0
$512$	22.8302	1.00896
$513$	3.11383	0.137479
$514$	3.77234	0.166391
$515$	−8.99656	−0.396436
$516$	45.5760	2.00637
$517$	19.3078	0.849155
$518$	0	0
$519$	−52.4070	−2.30041
$520$	−0.941367	−0.0412817
$521$	−43.9018	−1.92337	−0.961687	−	0.274149i	$-0.911604\pi$
$521$	−43.9018	−1.92337	−0.961687	+	0.274149i	$0.911604\pi$
$522$	5.05949	0.221448
$523$	−37.4328	−1.63682	−0.818410	−	0.574634i	$-0.805144\pi$
$523$	−37.4328	−1.63682	−0.818410	+	0.574634i	$0.805144\pi$
$524$	18.8793	0.824746
$525$	0	0
$526$	0.754297	0.0328889
$527$	9.19051	0.400345
$528$	13.7586	0.598766
$529$	11.0716	0.481375
$530$	−2.80949	−0.122037
$531$	25.0586	1.08745
$532$	0	0
$533$	6.49828	0.281472
$534$	7.31551	0.316573
$535$	2.94137	0.127166
$536$	28.3449	1.22431
$537$	−47.3269	−2.04231
$538$	5.56379	0.239872
$539$	0	0
$540$	1.99312	0.0857704
$541$	−34.9751	−1.50370	−0.751848	−	0.659336i	$-0.770837\pi$
$541$	−34.9751	−1.50370	−0.751848	+	0.659336i	$0.770837\pi$
$542$	−10.2998	−0.442416
$543$	37.6673	1.61646
$544$	6.32582	0.271217
$545$	−4.19395	−0.179649
$546$	0	0
$547$	6.50783	0.278255	0.139127	−	0.990274i	$-0.455570\pi$
$547$	6.50783	0.278255	0.139127	+	0.990274i	$0.455570\pi$
$548$	−20.2086	−0.863267
$549$	4.11727	0.175721
$550$	4.99656	0.213054
$551$	7.67924	0.327146
$552$	23.3484	0.993772
$553$	0	0
$554$	−4.80949	−0.204336
$555$	−2.81722	−0.119585
$556$	−24.7620	−1.05014
$557$	−43.4328	−1.84031	−0.920153	−	0.391559i	$-0.871936\pi$
$557$	−43.4328	−1.84031	−0.920153	+	0.391559i	$0.871936\pi$
$558$	6.80949	0.288269
$559$	−11.3940	−0.481915
$560$	0	0
$561$	6.61555	0.279309
$562$	0.725938	0.0306218
$563$	33.8827	1.42799	0.713993	−	0.700152i	$-0.246885\pi$
$563$	33.8827	1.42799	0.713993	+	0.700152i	$0.246885\pi$
$564$	−34.3380	−1.44589
$565$	5.23615	0.220287
$566$	−7.45875	−0.313515
$567$	0	0
$568$	−2.11727	−0.0888385
$569$	19.2147	0.805521	0.402760	−	0.915305i	$-0.368051\pi$
$569$	19.2147	0.805521	0.402760	+	0.915305i	$0.368051\pi$
$570$	0.824101	0.0345178
$571$	20.8268	0.871573	0.435787	−	0.900050i	$-0.356470\pi$
$571$	20.8268	0.871573	0.435787	+	0.900050i	$0.356470\pi$
$572$	−4.00000	−0.167248
$573$	16.7328	0.699023
$574$	0	0
$575$	−27.5500	−1.14892
$576$	−6.51127	−0.271303
$577$	−28.6448	−1.19250	−0.596249	−	0.802800i	$-0.703343\pi$
$577$	−28.6448	−1.19250	−0.596249	+	0.802800i	$0.703343\pi$
$578$	−7.19662	−0.299340
$579$	3.37758	0.140367
$580$	4.91539	0.204100
$581$	0	0
$582$	−3.67418	−0.152300
$583$	−25.3630	−1.05043
$584$	13.5930	0.562483
$585$	1.08967	0.0450523
$586$	−5.21649	−0.215491
$587$	4.32076	0.178337	0.0891685	−	0.996017i	$-0.471579\pi$
$587$	4.32076	0.178337	0.0891685	+	0.996017i	$0.471579\pi$
$588$	0	0
$589$	10.3354	0.425862
$590$	−3.03265	−0.124852
$591$	−53.9639	−2.21978
$592$	−6.43621	−0.264527
$593$	15.9690	0.655767	0.327883	−	0.944718i	$-0.393665\pi$
$593$	15.9690	0.655767	0.327883	+	0.944718i	$0.393665\pi$
$594$	−2.24141	−0.0919661
$595$	0	0
$596$	−16.5535	−0.678057
$597$	−4.53114	−0.185447
$598$	−2.74742	−0.112350
$599$	−16.8697	−0.689279	−0.344640	−	0.938735i	$-0.611999\pi$
$599$	−16.8697	−0.689279	−0.344640	+	0.938735i	$0.611999\pi$
$600$	−18.8793	−0.770744
$601$	15.3415	0.625792	0.312896	−	0.949787i	$-0.398701\pi$
$601$	15.3415	0.625792	0.312896	+	0.949787i	$0.398701\pi$
$602$	0	0
$603$	−32.8103	−1.33614
$604$	−12.5795	−0.511851
$605$	3.14486	0.127857
$606$	8.20532	0.333318
$607$	−35.8353	−1.45451	−0.727254	−	0.686368i	$-0.759204\pi$
$607$	−35.8353	−1.45451	−0.727254	+	0.686368i	$0.759204\pi$
$608$	7.11383	0.288504
$609$	0	0
$610$	−0.498281	−0.0201748
$611$	8.58451	0.347292
$612$	−4.78801	−0.193544
$613$	19.6673	0.794355	0.397177	−	0.917742i	$-0.369990\pi$
$613$	19.6673	0.794355	0.397177	+	0.917742i	$0.369990\pi$
$614$	−9.60761	−0.387732
$615$	−7.73625	−0.311956
$616$	0	0
$617$	−41.4588	−1.66907	−0.834533	−	0.550958i	$-0.814263\pi$
$617$	−41.4588	−1.66907	−0.834533	+	0.550958i	$0.814263\pi$
$618$	−17.9931	−0.723790
$619$	−10.8793	−0.437276	−0.218638	−	0.975806i	$-0.570161\pi$
$619$	−10.8793	−0.437276	−0.218638	+	0.975806i	$0.570161\pi$
$620$	6.61555	0.265687
$621$	12.3587	0.495937
$622$	0.905275	0.0362982
$623$	0	0
$624$	6.11727	0.244887
$625$	20.8759	0.835034
$626$	6.76203	0.270265
$627$	7.43965	0.297111
$628$	−10.7429	−0.428689
$629$	−3.09472	−0.123395
$630$	0	0
$631$	−31.4396	−1.25159	−0.625796	−	0.779987i	$-0.715226\pi$
$631$	−31.4396	−1.25159	−0.625796	+	0.779987i	$0.715226\pi$
$632$	2.38101	0.0947117
$633$	−22.7182	−0.902968
$634$	7.32238	0.290809
$635$	−0.436210	−0.0173105
$636$	45.1070	1.78861
$637$	0	0
$638$	−5.52770	−0.218844
$639$	2.45082	0.0969529
$640$	5.90871	0.233562
$641$	3.04221	0.120160	0.0600799	−	0.998194i	$-0.480864\pi$
$641$	3.04221	0.120160	0.0600799	+	0.998194i	$0.480864\pi$
$642$	5.88273	0.232173
$643$	−8.02922	−0.316641	−0.158321	−	0.987388i	$-0.550608\pi$
$643$	−8.02922	−0.316641	−0.158321	+	0.987388i	$0.550608\pi$
$644$	0	0
$645$	13.5646	0.534107
$646$	0.905275	0.0356176
$647$	−7.07324	−0.278078	−0.139039	−	0.990287i	$-0.544401\pi$
$647$	−7.07324	−0.278078	−0.139039	+	0.990287i	$0.544401\pi$
$648$	19.4526	0.764172
$649$	−27.3776	−1.07466
$650$	2.22154	0.0871361
$651$	0	0
$652$	−11.3484	−0.444436
$653$	−15.7586	−0.616681	−0.308341	−	0.951276i	$-0.599774\pi$
$653$	−15.7586	−0.616681	−0.308341	+	0.951276i	$0.599774\pi$
$654$	−8.38789	−0.327992
$655$	5.61899	0.219552
$656$	−17.6742	−0.690061
$657$	−15.7344	−0.613859
$658$	0	0
$659$	−12.2181	−0.475950	−0.237975	−	0.971271i	$-0.576484\pi$
$659$	−12.2181	−0.475950	−0.237975	+	0.971271i	$0.576484\pi$
$660$	4.76203	0.185362
$661$	−3.73443	−0.145253	−0.0726263	−	0.997359i	$-0.523138\pi$
$661$	−3.73443	−0.145253	−0.0726263	+	0.997359i	$0.523138\pi$
$662$	14.8501	0.577165
$663$	2.94137	0.114233
$664$	−29.0777	−1.12844
$665$	0	0
$666$	−2.29296	−0.0888506
$667$	30.4786	1.18014
$668$	−29.4948	−1.14119
$669$	22.8095	0.881866
$670$	3.97078	0.153405
$671$	−4.49828	−0.173654
$672$	0	0
$673$	−5.65775	−0.218090	−0.109045	−	0.994037i	$-0.534779\pi$
$673$	−5.65775	−0.218090	−0.109045	+	0.994037i	$0.534779\pi$
$674$	−3.96629	−0.152776
$675$	−9.99312	−0.384636
$676$	−1.77846	−0.0684022
$677$	−9.39906	−0.361235	−0.180618	−	0.983553i	$-0.557810\pi$
$677$	−9.39906	−0.361235	−0.180618	+	0.983553i	$0.557810\pi$
$678$	10.4723	0.402186
$679$	0	0
$680$	1.23109	0.0472103
$681$	−12.1104	−0.464071
$682$	−7.43965	−0.284879
$683$	−20.7328	−0.793319	−0.396660	−	0.917966i	$-0.629831\pi$
$683$	−20.7328	−0.793319	−0.396660	+	0.917966i	$0.629831\pi$
$684$	−5.38445	−0.205880
$685$	−6.01461	−0.229806
$686$	0	0
$687$	7.47250	0.285094
$688$	30.9897	1.18147
$689$	−11.2767	−0.429610
$690$	3.27083	0.124518
$691$	−16.0862	−0.611949	−0.305975	−	0.952040i	$-0.598982\pi$
$691$	−16.0862	−0.611949	−0.305975	+	0.952040i	$0.598982\pi$
$692$	−41.4396	−1.57530
$693$	0	0
$694$	9.10695	0.345695
$695$	−7.36984	−0.279554
$696$	20.8862	0.791688
$697$	−8.49828	−0.321895
$698$	−12.8286	−0.485570
$699$	−30.8578	−1.16715
$700$	0	0
$701$	−6.98013	−0.263636	−0.131818	−	0.991274i	$-0.542081\pi$
$701$	−6.98013	−0.263636	−0.131818	+	0.991274i	$0.542081\pi$
$702$	−0.996562	−0.0376128
$703$	−3.48024	−0.131260
$704$	7.11383	0.268112
$705$	−10.2199	−0.384905
$706$	12.0844	0.454803
$707$	0	0
$708$	48.6898	1.82988
$709$	8.39239	0.315183	0.157591	−	0.987504i	$-0.449627\pi$
$709$	8.39239	0.315183	0.157591	+	0.987504i	$0.449627\pi$
$710$	−0.296604	−0.0111313
$711$	−2.75612	−0.103362
$712$	12.2897	0.460577
$713$	41.0207	1.53624
$714$	0	0
$715$	−1.19051	−0.0445225
$716$	−37.4227	−1.39855
$717$	7.87586	0.294129
$718$	−11.0225	−0.411358
$719$	5.16129	0.192484	0.0962418	−	0.995358i	$-0.469318\pi$
$719$	5.16129	0.192484	0.0962418	+	0.995358i	$0.469318\pi$
$720$	−2.96371	−0.110451
$721$	0	0
$722$	−7.92494	−0.294936
$723$	3.57152	0.132826
$724$	29.7846	1.10693
$725$	−24.6448	−0.915284
$726$	6.28973	0.233434
$727$	40.4362	1.49970	0.749848	−	0.661610i	$-0.230127\pi$
$727$	40.4362	1.49970	0.749848	+	0.661610i	$0.230127\pi$
$728$	0	0
$729$	−8.23109	−0.304855
$730$	1.90422	0.0704782
$731$	14.9008	0.551125
$732$	8.00000	0.295689
$733$	39.1311	1.44534	0.722670	−	0.691193i	$-0.242915\pi$
$733$	39.1311	1.44534	0.722670	+	0.691193i	$0.242915\pi$
$734$	6.91215	0.255132
$735$	0	0
$736$	28.2345	1.04074
$737$	35.8466	1.32043
$738$	−6.29660	−0.231781
$739$	−7.13531	−0.262477	−0.131238	−	0.991351i	$-0.541895\pi$
$739$	−7.13531	−0.262477	−0.131238	+	0.991351i	$0.541895\pi$
$740$	−2.22766	−0.0818903
$741$	3.30777	0.121514
$742$	0	0
$743$	13.8827	0.509308	0.254654	−	0.967032i	$-0.418038\pi$
$743$	13.8827	0.509308	0.254654	+	0.967032i	$0.418038\pi$
$744$	28.1104	1.03058
$745$	−4.92676	−0.180502
$746$	−11.1398	−0.407857
$747$	33.6586	1.23150
$748$	5.23109	0.191268
$749$	0	0
$750$	−5.44652	−0.198879
$751$	−37.3251	−1.36201	−0.681005	−	0.732278i	$-0.738457\pi$
$751$	−37.3251	−1.36201	−0.681005	+	0.732278i	$0.738457\pi$
$752$	−23.3484	−0.851427
$753$	−11.0810	−0.403813
$754$	−2.45769	−0.0895039
$755$	−3.74398	−0.136258
$756$	0	0
$757$	−7.10428	−0.258209	−0.129105	−	0.991631i	$-0.541210\pi$
$757$	−7.10428	−0.258209	−0.129105	+	0.991631i	$0.541210\pi$
$758$	−15.4104	−0.559732
$759$	29.5277	1.07179
$760$	1.38445	0.0502194
$761$	25.9621	0.941125	0.470562	−	0.882367i	$-0.344051\pi$
$761$	25.9621	0.941125	0.470562	+	0.882367i	$0.344051\pi$
$762$	−0.872420	−0.0316044
$763$	0	0
$764$	13.2311	0.478684
$765$	−1.42504	−0.0515224
$766$	10.6448	0.384611
$767$	−12.1725	−0.439522
$768$	−2.41023	−0.0869717
$769$	21.4638	0.774005	0.387002	−	0.922079i	$-0.373511\pi$
$769$	21.4638	0.774005	0.387002	+	0.922079i	$0.373511\pi$
$770$	0	0
$771$	−18.0260	−0.649190
$772$	2.67074	0.0961221
$773$	40.4914	1.45637	0.728187	−	0.685378i	$-0.240363\pi$
$773$	40.4914	1.45637	0.728187	+	0.685378i	$0.240363\pi$
$774$	11.0404	0.396838
$775$	−33.1690	−1.19147
$776$	−6.17246	−0.221578
$777$	0	0
$778$	17.9087	0.642058
$779$	−9.55691	−0.342412
$780$	2.11727	0.0758103
$781$	−2.67762	−0.0958127
$782$	3.59301	0.128486
$783$	11.0554	0.395088
$784$	0	0
$785$	−3.19738	−0.114119
$786$	11.2380	0.400845
$787$	5.02072	0.178969	0.0894847	−	0.995988i	$-0.471478\pi$
$787$	5.02072	0.178969	0.0894847	+	0.995988i	$0.471478\pi$
$788$	−42.6707	−1.52008
$789$	−3.60438	−0.128319
$790$	0.333552	0.0118672
$791$	0	0
$792$	8.23453	0.292601
$793$	−2.00000	−0.0710221
$794$	5.50945	0.195523
$795$	13.4250	0.476137
$796$	−3.58289	−0.126992
$797$	−19.7002	−0.697815	−0.348908	−	0.937157i	$-0.613447\pi$
$797$	−19.7002	−0.697815	−0.348908	+	0.937157i	$0.613447\pi$
$798$	0	0
$799$	−11.2266	−0.397169
$800$	−22.8302	−0.807170
$801$	−14.2258	−0.502645
$802$	1.67418	0.0591174
$803$	17.1905	0.606640
$804$	−63.7517	−2.24835
$805$	0	0
$806$	−3.30777	−0.116511
$807$	−26.5863	−0.935883
$808$	13.7846	0.484940
$809$	−2.57678	−0.0905947	−0.0452974	−	0.998974i	$-0.514424\pi$
$809$	−2.57678	−0.0905947	−0.0452974	+	0.998974i	$0.514424\pi$
$810$	2.72508	0.0957496
$811$	−41.2311	−1.44782	−0.723910	−	0.689895i	$-0.757657\pi$
$811$	−41.2311	−1.44782	−0.723910	+	0.689895i	$0.757657\pi$
$812$	0	0
$813$	49.2173	1.72613
$814$	2.50516	0.0878057
$815$	−3.37758	−0.118311
$816$	−8.00000	−0.280056
$817$	16.7570	0.586252
$818$	−2.47680	−0.0865992
$819$	0	0
$820$	−6.11727	−0.213624
$821$	5.26719	0.183826	0.0919130	−	0.995767i	$-0.470702\pi$
$821$	5.26719	0.183826	0.0919130	+	0.995767i	$0.470702\pi$
$822$	−12.0292	−0.419567
$823$	36.0191	1.25555	0.627774	−	0.778396i	$-0.283966\pi$
$823$	36.0191	1.25555	0.627774	+	0.778396i	$0.283966\pi$
$824$	−30.2277	−1.05303
$825$	−23.8759	−0.831251
$826$	0	0
$827$	−16.3157	−0.567353	−0.283676	−	0.958920i	$-0.591554\pi$
$827$	−16.3157	−0.567353	−0.283676	+	0.958920i	$0.591554\pi$
$828$	−21.3707	−0.742683
$829$	30.3956	1.05568	0.527842	−	0.849343i	$-0.323001\pi$
$829$	30.3956	1.05568	0.527842	+	0.849343i	$0.323001\pi$
$830$	−4.07344	−0.141391
$831$	22.9820	0.797235
$832$	3.16291	0.109654
$833$	0	0
$834$	−14.7397	−0.510394
$835$	−8.77846	−0.303791
$836$	5.88273	0.203459
$837$	14.8793	0.514304
$838$	−12.2536	−0.423295
$839$	−29.8398	−1.03018	−0.515092	−	0.857135i	$-0.672242\pi$
$839$	−29.8398	−1.03018	−0.515092	+	0.857135i	$0.672242\pi$
$840$	0	0
$841$	−1.73549	−0.0598445
$842$	10.4691	0.360788
$843$	−3.46886	−0.119474
$844$	−17.9639	−0.618343
$845$	−0.529317	−0.0182090
$846$	−8.31809	−0.285982
$847$	0	0
$848$	30.6707	1.05324
$849$	35.6413	1.22321
$850$	−2.90528	−0.0996501
$851$	−13.8129	−0.473501
$852$	4.76203	0.163144
$853$	0.203497	0.00696761	0.00348380	−	0.999994i	$-0.498891\pi$
$853$	0.203497	0.00696761	0.00348380	+	0.999994i	$0.498891\pi$
$854$	0	0
$855$	−1.60256	−0.0548063
$856$	9.88273	0.337785
$857$	12.6155	0.430939	0.215469	−	0.976511i	$-0.430872\pi$
$857$	12.6155	0.430939	0.215469	+	0.976511i	$0.430872\pi$
$858$	−2.38101	−0.0812865
$859$	27.9671	0.954227	0.477113	−	0.878842i	$-0.341683\pi$
$859$	27.9671	0.954227	0.477113	+	0.878842i	$0.341683\pi$
$860$	10.7259	0.365751
$861$	0	0
$862$	13.0258	0.443660
$863$	−2.76891	−0.0942546	−0.0471273	−	0.998889i	$-0.515007\pi$
$863$	−2.76891	−0.0942546	−0.0471273	+	0.998889i	$0.515007\pi$
$864$	10.2414	0.348420
$865$	−12.3336	−0.419353
$866$	6.01910	0.204537
$867$	34.3887	1.16790
$868$	0	0
$869$	3.01117	0.102147
$870$	2.92590	0.0991974
$871$	15.9379	0.540036
$872$	−14.0913	−0.477191
$873$	7.14486	0.241817
$874$	4.04059	0.136675
$875$	0	0
$876$	−30.5726	−1.03295
$877$	6.71133	0.226626	0.113313	−	0.993559i	$-0.463854\pi$
$877$	6.71133	0.226626	0.113313	+	0.993559i	$0.463854\pi$
$878$	8.55024	0.288557
$879$	24.9268	0.840759
$880$	3.23797	0.109152
$881$	18.3741	0.619040	0.309520	−	0.950893i	$-0.399832\pi$
$881$	18.3741	0.619040	0.309520	+	0.950893i	$0.399832\pi$
$882$	0	0
$883$	9.93105	0.334207	0.167103	−	0.985939i	$-0.446559\pi$
$883$	9.93105	0.334207	0.167103	+	0.985939i	$0.446559\pi$
$884$	2.32582	0.0782258
$885$	14.4914	0.487123
$886$	0.0506994	0.00170328
$887$	51.3776	1.72509	0.862545	−	0.505980i	$-0.168869\pi$
$887$	51.3776	1.72509	0.862545	+	0.505980i	$0.168869\pi$
$888$	−9.46563	−0.317646
$889$	0	0
$890$	1.72164	0.0577096
$891$	24.6009	0.824162
$892$	18.0361	0.603893
$893$	−12.6251	−0.422483
$894$	−9.85352	−0.329551
$895$	−11.1380	−0.372302
$896$	0	0
$897$	13.1284	0.438346
$898$	−10.4362	−0.348261
$899$	36.6949	1.22384
$900$	17.2802	0.576006
$901$	14.7474	0.491308
$902$	6.87930	0.229055
$903$	0	0
$904$	17.5930	0.585135
$905$	8.86469	0.294672
$906$	−7.48797	−0.248771
$907$	4.34225	0.144182	0.0720910	−	0.997398i	$-0.477033\pi$
$907$	4.34225	0.144182	0.0720910	+	0.997398i	$0.477033\pi$
$908$	−9.57602	−0.317791
$909$	−15.9562	−0.529233
$910$	0	0
$911$	31.4853	1.04315	0.521577	−	0.853204i	$-0.325344\pi$
$911$	31.4853	1.04315	0.521577	+	0.853204i	$0.325344\pi$
$912$	−8.99656	−0.297906
$913$	−36.7734	−1.21702
$914$	2.05196	0.0678728
$915$	2.38101	0.0787139
$916$	5.90871	0.195229
$917$	0	0
$918$	1.30328	0.0430146
$919$	43.4588	1.43357	0.716786	−	0.697293i	$-0.245612\pi$
$919$	43.4588	1.43357	0.716786	+	0.697293i	$0.245612\pi$
$920$	5.49484	0.181160
$921$	45.9096	1.51277
$922$	−15.2412	−0.501942
$923$	−1.19051	−0.0391860
$924$	0	0
$925$	11.1690	0.367235
$926$	3.93793	0.129408
$927$	34.9897	1.14921
$928$	25.2571	0.829104
$929$	−4.40517	−0.144529	−0.0722645	−	0.997385i	$-0.523023\pi$
$929$	−4.40517	−0.144529	−0.0722645	+	0.997385i	$0.523023\pi$
$930$	3.93793	0.129130
$931$	0	0
$932$	−24.4001	−0.799252
$933$	−4.32582	−0.141621
$934$	9.19824	0.300976
$935$	1.55691	0.0509165
$936$	3.66119	0.119670
$937$	34.0990	1.11397	0.556983	−	0.830524i	$-0.311959\pi$
$937$	34.0990	1.11397	0.556983	+	0.830524i	$0.311959\pi$
$938$	0	0
$939$	−32.3121	−1.05446
$940$	−8.08117	−0.263579
$941$	−44.4672	−1.44959	−0.724795	−	0.688964i	$-0.758066\pi$
$941$	−44.4672	−1.44959	−0.724795	+	0.688964i	$0.758066\pi$
$942$	−6.39477	−0.208353
$943$	−37.9311	−1.23521
$944$	33.1070	1.07754
$945$	0	0
$946$	−12.0621	−0.392172
$947$	57.9311	1.88251	0.941253	−	0.337702i	$-0.109650\pi$
$947$	57.9311	1.88251	0.941253	+	0.337702i	$0.109650\pi$
$948$	−5.35524	−0.173930
$949$	7.64315	0.248107
$950$	−3.26719	−0.106002
$951$	−34.9897	−1.13462
$952$	0	0
$953$	35.3060	1.14367	0.571836	−	0.820368i	$-0.306231\pi$
$953$	35.3060	1.14367	0.571836	+	0.820368i	$0.306231\pi$
$954$	10.9268	0.353767
$955$	3.93793	0.127428
$956$	6.22766	0.201417
$957$	26.4139	0.853839
$958$	13.4250	0.433743
$959$	0	0
$960$	−3.76547	−0.121530
$961$	18.3871	0.593133
$962$	1.11383	0.0359113
$963$	−11.4396	−0.368638
$964$	2.82410	0.0909582
$965$	0.794885	0.0255882
$966$	0	0
$967$	23.7148	0.762616	0.381308	−	0.924448i	$-0.375474\pi$
$967$	23.7148	0.762616	0.381308	+	0.924448i	$0.375474\pi$
$968$	10.5665	0.339619
$969$	−4.32582	−0.138965
$970$	−0.864688	−0.0277635
$971$	23.9379	0.768205	0.384102	−	0.923291i	$-0.374511\pi$
$971$	23.9379	0.768205	0.384102	+	0.923291i	$0.374511\pi$
$972$	−32.4553	−1.04100
$973$	0	0
$974$	11.6843	0.374389
$975$	−10.6155	−0.339970
$976$	5.43965	0.174119
$977$	−16.1871	−0.517870	−0.258935	−	0.965895i	$-0.583372\pi$
$977$	−16.1871	−0.517870	−0.258935	+	0.965895i	$0.583372\pi$
$978$	−6.75515	−0.216006
$979$	15.5423	0.496734
$980$	0	0
$981$	16.3112	0.520777
$982$	13.7294	0.438122
$983$	−45.7243	−1.45838	−0.729190	−	0.684312i	$-0.760103\pi$
$983$	−45.7243	−1.45838	−0.729190	+	0.684312i	$0.760103\pi$
$984$	−25.9931	−0.828631
$985$	−12.7000	−0.404654
$986$	3.21411	0.102358
$987$	0	0
$988$	2.61555	0.0832116
$989$	66.5078	2.11483
$990$	1.15356	0.0366625
$991$	−5.40356	−0.171650	−0.0858248	−	0.996310i	$-0.527353\pi$
$991$	−5.40356	−0.171650	−0.0858248	+	0.996310i	$0.527353\pi$
$992$	33.9931	1.07928
$993$	−70.9605	−2.25186
$994$	0	0
$995$	−1.06637	−0.0338061
$996$	65.3999	2.07228
$997$	−6.04832	−0.191552	−0.0957761	−	0.995403i	$-0.530533\pi$
$997$	−6.04832	−0.191552	−0.0957761	+	0.995403i	$0.530533\pi$
$998$	−15.6742	−0.496158
$999$	−5.01031	−0.158519

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	637.2.a.j.1.2		3
3.2	odd	2		5733.2.a.x.1.2		3
7.2	even	3		637.2.e.i.508.2		6
7.3	odd	6		637.2.e.j.79.2		6
7.4	even	3		637.2.e.i.79.2		6
7.5	odd	6		637.2.e.j.508.2		6
7.6	odd	2		91.2.a.d.1.2	✓	3
13.12	even	2		8281.2.a.bg.1.2		3
21.20	even	2		819.2.a.i.1.2		3
28.27	even	2		1456.2.a.t.1.1		3
35.34	odd	2		2275.2.a.m.1.2		3
56.13	odd	2		5824.2.a.by.1.1		3
56.27	even	2		5824.2.a.bs.1.3		3
91.34	even	4		1183.2.c.f.337.4		6
91.83	even	4		1183.2.c.f.337.3		6
91.90	odd	2		1183.2.a.i.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.a.d.1.2	✓	3	7.6	odd	2
637.2.a.j.1.2		3	1.1	even	1	trivial
637.2.e.i.79.2		6	7.4	even	3
637.2.e.i.508.2		6	7.2	even	3
637.2.e.j.79.2		6	7.3	odd	6
637.2.e.j.508.2		6	7.5	odd	6
819.2.a.i.1.2		3	21.20	even	2
1183.2.a.i.1.2		3	91.90	odd	2
1183.2.c.f.337.3		6	91.83	even	4
1183.2.c.f.337.4		6	91.34	even	4
1456.2.a.t.1.1		3	28.27	even	2
2275.2.a.m.1.2		3	35.34	odd	2
5733.2.a.x.1.2		3	3.2	odd	2
5824.2.a.bs.1.3		3	56.27	even	2
5824.2.a.by.1.1		3	56.13	odd	2
8281.2.a.bg.1.2		3	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.a (trivial)

\(f(q)\)	\(=\)	\(q+0.470683 q^{2} -2.24914 q^{3} -1.77846 q^{4} -0.529317 q^{5} -1.05863 q^{6} -1.77846 q^{8} +2.05863 q^{9} -0.249141 q^{10} -2.24914 q^{11} +4.00000 q^{12} -1.00000 q^{13} +1.19051 q^{15} +2.71982 q^{16} +1.30777 q^{17} +0.968964 q^{18} +1.47068 q^{19} +0.941367 q^{20} -1.05863 q^{22} +5.83709 q^{23} +4.00000 q^{24} -4.71982 q^{25} -0.470683 q^{26} +2.11727 q^{27} +5.22154 q^{29} +0.560352 q^{30} +7.02760 q^{31} +4.83709 q^{32} +5.05863 q^{33} +0.615547 q^{34} -3.66119 q^{36} -2.36641 q^{37} +0.692226 q^{38} +2.24914 q^{39} +0.941367 q^{40} -6.49828 q^{41} +11.3940 q^{43} +4.00000 q^{44} -1.08967 q^{45} +2.74742 q^{46} -8.58451 q^{47} -6.11727 q^{48} -2.22154 q^{50} -2.94137 q^{51} +1.77846 q^{52} +11.2767 q^{53} +0.996562 q^{54} +1.19051 q^{55} -3.30777 q^{57} +2.45769 q^{58} +12.1725 q^{59} -2.11727 q^{60} +2.00000 q^{61} +3.30777 q^{62} -3.16291 q^{64} +0.529317 q^{65} +2.38101 q^{66} -15.9379 q^{67} -2.32582 q^{68} -13.1284 q^{69} +1.19051 q^{71} -3.66119 q^{72} -7.64315 q^{73} -1.11383 q^{74} +10.6155 q^{75} -2.61555 q^{76} +1.05863 q^{78} -1.33881 q^{79} -1.43965 q^{80} -10.9379 q^{81} -3.05863 q^{82} +16.3500 q^{83} -0.692226 q^{85} +5.36297 q^{86} -11.7440 q^{87} +4.00000 q^{88} -6.91033 q^{89} -0.512889 q^{90} -10.3810 q^{92} -15.8061 q^{93} -4.04059 q^{94} -0.778457 q^{95} -10.8793 q^{96} +3.47068 q^{97} -4.63016 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 2 q^{3} + 3 q^{4} - 2 q^{5} - 4 q^{6} + 3 q^{8} + 7 q^{9} + 8 q^{10} + 2 q^{11} + 12 q^{12} - 3 q^{13} - 6 q^{15} - q^{16} - 4 q^{17} - 15 q^{18} + 4 q^{19} + 2 q^{20} - 4 q^{22} + 10 q^{23}+ \cdots + 14 q^{99}+O(q^{100}) \) 3 * q + q^2 + 2 * q^3 + 3 * q^4 - 2 * q^5 - 4 * q^6 + 3 * q^8 + 7 * q^9 + 8 * q^10 + 2 * q^11 + 12 * q^12 - 3 * q^13 - 6 * q^15 - q^16 - 4 * q^17 - 15 * q^18 + 4 * q^19 + 2 * q^20 - 4 * q^22 + 10 * q^23 + 12 * q^24 - 5 * q^25 - q^26 + 8 * q^27 + 24 * q^29 + 20 * q^30 + 4 * q^31 + 7 * q^32 + 16 * q^33 - 14 * q^34 - q^36 + 10 * q^38 - 2 * q^39 + 2 * q^40 - 2 * q^41 + 10 * q^43 + 12 * q^44 - 22 * q^45 - 18 * q^46 + 8 * q^47 - 20 * q^48 - 15 * q^50 - 8 * q^51 - 3 * q^52 + 8 * q^53 - 32 * q^54 - 6 * q^55 - 2 * q^57 + 12 * q^58 + 4 * q^59 - 8 * q^60 + 6 * q^61 + 2 * q^62 - 17 * q^64 + 2 * q^65 - 12 * q^66 - 12 * q^67 - 22 * q^68 + 6 * q^69 - 6 * q^71 - q^72 + 10 * q^73 + 30 * q^74 + 16 * q^75 + 8 * q^76 + 4 * q^78 - 14 * q^79 + 14 * q^80 + 3 * q^81 - 10 * q^82 + 12 * q^83 - 10 * q^85 - 26 * q^86 + 26 * q^87 + 12 * q^88 - 2 * q^89 + 28 * q^90 - 12 * q^92 - 22 * q^93 + 10 * q^94 + 6 * q^95 + 4 * q^96 + 10 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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